DC motor modeling and control loop design

Motor description

Motor Scheme
Variables and parameters table

System modeling

Electric equation:

From kirchhoff's voltage law :

Ldi(t)dt=v(t)Ri(t)e(t)L\frac{di(t)}{dt} =v(t) - Ri(t) - e(t)

Mechanical equation:

By Newton's law:

Jdω(t)dt=T(t)=Tm(t)fω(t)J\frac{d\omega(t)}{dt} = \sum T(t) = T_m(t)-f\omega(t)

where T(t)T(t) is the total torque applied on the rotor.

Electro-mechanical coupling:

The back EMF is proportional to speed :

e(t)=Keω(t)e(t) = K_e \omega(t)

KeK_e : electromotive constant (V.rad1.s)(V.rad^{-1}.s)

The motor torque is proportional to current :

Tm(t)=KTi(t)T_m(t) = K_T i(t)

KTK_T : Torque constant (N.m.A1)(N.m.A^{-1})

The mechanical power produced by the DC motor is Tmω=KTiωT_m\omega = K_Ti\omega. The electric power Pe=viP_e = vi delivered by the source goes into heat loss in the resistance RR, into stored magnetic energy in the inductance LL and the remaining quantity iKeωiK_e\omega is converted in mechanical energy TmωT_m\omega. It leads to Tmω=KTiω=KeiωT_m\omega = K_Ti\omega = K_ei\omega whether KT=Ke=KϕK_T = K_e = K_\phi (Chiasson2005).

Motor Bloc Diagram

DC Motor Bloc Diagram

Motor control : cascaded strategy

The control synthesis is inspired by Permanent Magnet Synchronous Motor control synthesis based on cascaded control synthesis. Due to frequency separation the control can be divided into two control loops. The inner loop control the electrical dynamic while the outer loop treats the mechanical dynamic. Generally the the electrical dynamics is neglected and the mechanical dynamics is considered only. However in the case where motor resistance is low, this strategy can damage the motor.

Electrical dynamics control

The objective is to control the motor torque Tm(t)T_m(t). Indeed Tm(t)=Kϕi(t)T_m(t) = K_\phi i(t) the motor torque is imposed by the current.

Closed loop electrical system

With the assumption that the mechanical dynamic is slower the the electrical one, one has :

τelec=LR<<τmeca=Jf\tau_{\rm elec} = \frac{L}{R}<<\tau_{\rm meca} = \frac{J}{f}

The velocity ω\omega can then be considered as constant from the point of view of the electrical dynamics.

Feedback control with integral action

The electrical dynamics is given by

i˙=RLi+1LvKϕLw=1τei+KeτevKϕLw\begin{array}{lcl} \dot{i} &=& -\frac{R}{L} i + \frac{1}{L}v -\frac{K_\phi}{L}w\\ &=& -\frac{1}{\tau_e} i + \frac{K_e}{\tau_e}v -\frac{K_\phi}{L}w \end{array}

The control objective is to ensure i=irefi^\star =i_{\rm ref}, where ii^\star is the current steady state and irefi_{\rm ref} is the current reference. To ensure zero steady state error, an integral action is necessary. The principle is to insert an integral action the the feed-forward loop between the error compactor and the process (Ogata2010). The control scheme is given by :

Electrical dynamics state feedback

From the figure one gets :

i˙=RLi+1LvKϕLwε˙=irefiv=Ki+KIε\begin{array}{lcl} \dot{i} & = & -\frac{R}{L} i + \frac{1}{L}v -\frac{K_\phi}{L}w \\ \dot{\varepsilon} & = & i_{\rm ref} - i \\ v & = & -Ki+K_I\varepsilon \end{array}

with ε\varepsilon the output of the integrator.

The system dynamics can be described by

[i˙ε˙]=[RL010][iε]+[1L0]v+[KϕL0]ω+[01]iref\begin{bmatrix} \dot{i}\\\dot{\varepsilon} \end{bmatrix} = \begin{bmatrix} -\frac{R}{L} & 0\\-1 &0 \end{bmatrix} \begin{bmatrix} {i}\\{\varepsilon} \end{bmatrix}+ \begin{bmatrix} \frac{1}{L}\\0 \end{bmatrix}v+ \begin{bmatrix} \frac{K_\phi}{L}\\0 \end{bmatrix}\omega+ \begin{bmatrix} 0\\1 \end{bmatrix}i_{\rm ref}

with the control

v=KiKIεv = -Ki-K_I\varepsilon

The closed loop system leads to :

[i˙ξ˙]=([RL010][1L0][KKI][iε])+[KϕL0]ω+[01]iref\begin{bmatrix} \dot{i}\\\dot{\xi} \end{bmatrix} = \left(\begin{bmatrix} -\frac{R}{L} & 0\\-1 &0 \end{bmatrix} - \begin{bmatrix} \frac{1}{L}\\0 \end{bmatrix} \begin{bmatrix}K&K_I\end{bmatrix}\begin{bmatrix}i\\\varepsilon\end{bmatrix}\right)+ \begin{bmatrix} \frac{K_\phi}{L}\\0 \end{bmatrix}\omega+ \begin{bmatrix} 0\\1 \end{bmatrix}i_{\rm ref}

The closed loop dynamics depends on the eigenvalues of the matrix :

([RL010][1L0][KKI])=[R+KLKIL10]\left(\begin{bmatrix} -\frac{R}{L} & 0\\-1 &0 \end{bmatrix} - \begin{bmatrix} \frac{1}{L}\\0 \end{bmatrix} \begin{bmatrix}K&K_I\end{bmatrix}\right)= \begin{bmatrix} -\frac{R+K}{L} & -\frac{K_I}{L}\\-1 &0 \end{bmatrix}

One has

eig([R+KLKIL10])=det(sI[R+KLKIL10]){\rm eig}\left( \begin{bmatrix} -\frac{R+K}{L} & -\frac{K_I}{L}\\-1 &0 \end{bmatrix}\right) = {\rm det}\left(sI-\begin{bmatrix} -\frac{R+K}{L} & -\frac{K_I}{L}\\-1 &0 \end{bmatrix}\right)

It leads to a characteristic equation

P(s)=s2+K+RLsKIP(s) = s^2+\frac{K+R}{L}s -K_I

to be identified with the classical second order characteristic equation

P(s)=p2+2ζωns+ωn2P(s) = p^2+2\zeta\omega_n s +\omega_n^2

where ωn\omega_n is the desired closed loop natural frequency and ζ\zeta the damping coefficient.

Mechanical dynamics control

Closed loop cascaded system

Assuming the electrical control has been correctly synthesized with respect to frequency separation principle, which means that the closed loop electrical dynamics is faster than the mechanical desired dynamics, then the mechanical dynamics control synthesis can be designed without considering the closed loop electrical system. The control scheme can be simplified as :

Closed loop mechanical system

The mechanical dynamics is

ω˙=1JTmfJω=KϕJifJω\begin{array}{lcl} \dot{\omega} &=& \frac{1}{J}T_m-\frac{f}{J}\omega\\ &=& \frac{K_\phi}{J}i-\frac{f}{J}\omega \end{array}

where Tm=KϕiT_m = K_\phi i

The control synthesis is similar than the one proposed for the electrical dynamics with ε˙ω=ωrefω\dot\varepsilon_\omega = \omega_{\rm ref}-\omega leading to

[ω˙εω˙]=[fJ010][ωεω]+[KϕJ0]i+[01]ωref\begin{bmatrix} \dot{\omega}\\\dot{\varepsilon_\omega} \end{bmatrix} = \begin{bmatrix} -\frac{f}{J} & 0\\-1 &0 \end{bmatrix} \begin{bmatrix} {\omega}\\{\varepsilon_\omega} \end{bmatrix}+ \begin{bmatrix} \frac{K_\phi}{J}\\0 \end{bmatrix}i+ \begin{bmatrix} 0\\1 \end{bmatrix}\omega_{\rm ref}

with the control

i=KωωKω,Iεωi = -K_\omega \omega-K_{\omega,I}\varepsilon_\omega

By analogy, it leads to a characteristic equation

P(s)=s2+KϕKω+fLsKϕKω,IP(s) = s^2+\frac{K_\phi K_\omega+f}{L}s -K\phi K_{\omega,I}

to be identified with the classical second order characteristic equation

P(s)=p2+2ζωns+ωn2.P(s) = p^2+2\zeta\omega_n s +\omega_n^2 .

References

(Chiasson2005) Chiasson, J.-N. (2005). Modeling and High-Performance Control of Electric Machines (IEEE Press).

(Ogata2010) Ogata, K. (2010). Modern Control Engineering. Prentice Hall.